Graph Learning Analyses

Tasks

Participants completed three tasks: 1.) Graph-learning, 2.) Graph-parsing, and 3.) Graph-reconstruction.

Learning task

In the learning task, on each trial, participants saw a series of planets with overlaid gabor patches. Participants had to press one of two keys to indicate the ‘direction of the wind’ on the planet, as indicated by the gabor. Wind directions were determined randomly on every trial.

Unbeknownst to participants, the order of the planets was determined by a random walk along a modular graph. Each planet transitioned with 25% probability to one of 4 other planets. The graph structure created three ‘communities’ of planets, and two types of transitions: within-community transitions, and between-community transitions. Participants completed a total of 600 learning trials.

Parsing task

After the learning task, participants completed a ‘parsing’ task, in which they once again saw a sequence of planets (following the same random walk) and had to press the space bar to indicate when they believed they had entered a different part of the galaxy.

Participants completed a total of 600 parsing trials.

Graph reconstruction

At the end of the experiment, participants saw all the planets placed randomly on the screen. They were told to drag them around the screen to create a map based on how they thought ‘the planets were spread out across the universe.’ Participants could click on each planet to move them around the screen, pressing a final button to submit their map when they were happy with it.

Format data for SR modeling

Apply participant exclusions

Participants were excluded from the learning task if they meet any of the following criteria: - More than 20 browser interactions - Less than 75% accuracy on learning trials (excluding missed trials) - More than 20% (120) of learning trials missed - More than 20% (120 of learning trials fast (< 200 ms)) - More than 30 minutes (1800000 ms) from start of last trial of block 1 to first trial of block 2

Table continues below

age_group	exclude_browser	exclude_missed_trials	exclude_acc
Children	1	10	16
Adolescents	3	1	5
Adults	3	3	2

exclude_fast_rts	exclude_long_break	total_excluded
1	0	28
0	2	11
0	2	10

age_group	N
Children	27
Adolescents	35
Adults	44

Results

Learning

For all learning analyses, we will examine only accurate trials in which participants responded in > 200 ms

Response time distributions

Response times by age, within block trial, block, planet lag, transition type

transition_type	mean_lag	sd_lag
between	20.67	25.48
within	10.75	19.9

Learning model with quadratic age

Plot: Log response times across planet lags

Plot: Response times across planet lags with transition type

Plot: Response times across trials with transition type

Parsing

Parsing responses by age, trial, transition type

Plot: Parsing responses by age group and transition type

Parsing responses by age, trial, transition type, planet lag

Children-only parsing responses by age, trial, transition type

Plot: Parsing responses by age, transition type, planet lag, trial

Plot: Probability of parsing as a functon of trials around transition types

Graph reconstruction

Filter out subjects who didn’t move the planets

Compute empirical distance matrices

Compute ground-truth geodesic distances

RSA with geodesic matrix

Is RSA above 0?

## 
##  One Sample t-test
## 
## data:  similarity_df$rsa
## t = 7.0617, df = 105, p-value = 1.852e-10
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  0.1025174 0.1825632
## sample estimates:
## mean of x 
## 0.1425403

RSA scores by age

## 
## Call:
## lm(formula = rsa ~ (age_z + age_squared_z), data = similarity_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.30127 -0.15114 -0.07387  0.15610  0.59975 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    0.14254    0.02014   7.078 1.84e-10 ***
## age_z          0.27425    0.18296   1.499    0.137    
## age_squared_z -0.26260    0.18296  -1.435    0.154    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2073 on 103 degrees of freedom
## Multiple R-squared:  0.0236, Adjusted R-squared:  0.00464 
## F-statistic: 1.245 on 2 and 103 DF,  p-value: 0.2923

Model-derived parameter analyses

Relations between parameters and age

Plot: Relation between β_C and age

Plot: Relation between β_R and age

Plot: Relation between γ and age

Plot: Relation between β_C and β_R

Relations between parameters and explicit knowledge

Relation between β_C and graph reconstruction

## 
## Call:
## lm(formula = rsa ~ beta_c_z, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.28046 -0.15767 -0.07212  0.15079  0.56821 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.142540   0.020274   7.031 2.23e-10 ***
## beta_c_z    -0.005775   0.020370  -0.283    0.777    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2087 on 104 degrees of freedom
## Multiple R-squared:  0.0007722,  Adjusted R-squared:  -0.008836 
## F-statistic: 0.08037 on 1 and 104 DF,  p-value: 0.7774

Relation between γ and graph reconstruction

## 
## Call:
## lm(formula = rsa ~ gamma_z, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.27636 -0.15880 -0.07043  0.15557  0.57753 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.142540   0.020281   7.028 2.26e-10 ***
## gamma_z     -0.001001   0.020378  -0.049    0.961    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2088 on 104 degrees of freedom
## Multiple R-squared:  2.32e-05,   Adjusted R-squared:  -0.009592 
## F-statistic: 0.002412 on 1 and 104 DF,  p-value: 0.9609

Relation between β_R and rsa

## 
## Call:
## lm(formula = rsa ~ beta_r_z, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.28199 -0.15849 -0.07236  0.15067  0.57379 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.142540   0.020260   7.036 2.18e-10 ***
## beta_r_z    -0.009689   0.020356  -0.476    0.635    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2086 on 104 degrees of freedom
## Multiple R-squared:  0.002174,   Adjusted R-squared:  -0.007421 
## F-statistic: 0.2265 on 1 and 104 DF,  p-value: 0.6351

Relation between β_C and parsing

## Mixed Model Anova Table (Type 3 tests, LRT-method)
## 
## Model: parse ~ (age_z + beta_c_z) * (transition_type + planet_lag_z + 
## Model:     within_block_trial_z + block_z) + (transition_type + planet_lag_z + 
## Model:     within_block_trial_z + block_z || subject_id)
## Data: parsing_parameter_mixed_model_data
## Df full model: 20
##                           Effect df     Chisq p.value
## 1                          age_z  1    5.57 *    .018
## 2                       beta_c_z  1      2.24    .134
## 3                transition_type  1 24.59 ***   <.001
## 4                   planet_lag_z  1 52.64 ***   <.001
## 5           within_block_trial_z  1    3.40 +    .065
## 6                        block_z  1    5.71 *    .017
## 7          age_z:transition_type  1      0.24    .625
## 8             age_z:planet_lag_z  1      1.57    .210
## 9     age_z:within_block_trial_z  1      1.56    .211
## 10                 age_z:block_z  1    3.86 *    .049
## 11      beta_c_z:transition_type  1      0.15    .695
## 12         beta_c_z:planet_lag_z  1      1.12    .289
## 13 beta_c_z:within_block_trial_z  1      1.67    .197
## 14              beta_c_z:block_z  1    2.90 +    .089
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

Relation between β_R and parsing

## Mixed Model Anova Table (Type 3 tests, LRT-method)
## 
## Model: parse ~ (age_z + beta_r_z) * (transition_type + planet_lag_z + 
## Model:     within_block_trial_z + block_z) + (transition_type + planet_lag_z + 
## Model:     within_block_trial_z + block_z || subject_id)
## Data: parsing_parameter_mixed_model_data
## Df full model: 20
##                           Effect df     Chisq p.value
## 1                          age_z  1   8.37 **    .004
## 2                       beta_r_z  1      0.04    .840
## 3                transition_type  1 25.40 ***   <.001
## 4                   planet_lag_z  1 52.98 ***   <.001
## 5           within_block_trial_z  1    3.25 +    .071
## 6                        block_z  1    5.55 *    .018
## 7          age_z:transition_type  1  10.75 **    .001
## 8             age_z:planet_lag_z  1      1.77    .183
## 9     age_z:within_block_trial_z  1      0.02    .895
## 10                 age_z:block_z  1      0.61    .435
## 11      beta_r_z:transition_type  1    2.80 +    .094
## 12         beta_r_z:planet_lag_z  1      1.79    .181
## 13 beta_r_z:within_block_trial_z  1      0.19    .660
## 14              beta_r_z:block_z  1      0.39    .534
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1